Conjunction iff Biconditional of Biconditional with Disjunction
Jump to navigation
Jump to search
Theorem
- $p \land q \dashv \vdash \paren {p \iff q} \iff \paren {p \lor q}$
Proof
We apply the Method of Truth Tables.
As can be seen by inspection, the appropriate truth values match for all boolean interpretations.
$\begin{array}{|ccc||ccccccc|} \hline p & \land & q & (p & \iff & q) & \iff & (p & \lor & q) \\ \hline \F & \F & \F & \F & \T & \F & \F & \F & \F & \F \\ \F & \F & \T & \F & \F & \T & \F & \F & \T & \T \\ \T & \F & \F & \T & \F & \F & \F & \T & \T & \F \\ \T & \T & \T & \T & \T & \T & \T & \T & \T & \T \\ \hline \end{array}$
$\blacksquare$
Sources
- 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.10$: Exercise $2.6$